Are there infinite sets $E\subset\mathbb{Z}$ such that any $f\in l^2(\mathbb{Z})$ with support on $E$ comes from the Fourier transform of a continuous function on $\mathbb{T}$ ? If yes, is there a characterization of these sets?

If we change the group from $\mathbb{Z}$ to the free group with n generators $\mathbb{F_n}$, $n\geq 2$,then an infinite set $E$ with the property that $x_{i_1}x_{i_2}^{-1}x_{i_3}x_{i_4}^{-1}\ldots x_{i_{2k-1}}x_{i_{2k}}^{-1}\neq 1,\ \forall k\in\mathbb{N},\ \forall x_{i_1}\neq x_{i_2},\ x_{i_2}\neq x_{i_3},\ldots x_{i_{2k-1}}\neq x_{2k}$ (studied by Leinert, Bozejko) will satisfy the following:

Any $f\in l^2(\mathbb{F_n})$ with support on $E$ will be in the reduced $C^*$ algebra $C^*_r(\mathbb{F_n})$, and moreover $$||f||_\infty\leq 2||f||_2$$

[Edit:] By $||\cdot||_\infty$ I mean the operator norm of the corresponding convolutor.